MATLAB 2D/3D Beamforming Simulation with Cross-Spectrum Analysis for Antenna Arrays

 

MATLAB Code

%% 2D Conceptual + Electronically Steered Beamforming

clc; clear; close all;

%% PARAMETERS

Fs = 1000; % Sampling frequency [Hz]

T = 1/Fs; % Sampling period

t = 0:T:1-T; % 1-second time vector

f_sig = 50; % Signal frequency [Hz]

%% SIMULATE ANTENNA SIGNALS

p = sin(2*pi*f_sig*t); % Reference signal (antenna p)

vx = 0.8*sin(2*pi*f_sig*t + pi/6); % vx antenna

vy = 0.6*sin(2*pi*f_sig*t + pi/4); % vy antenna

%% NORMALIZE SIGNALS

pnor = p / max(abs(p));

vxnor = vx / max(abs(vx));

vynor = vy / max(abs(vy));

%% PASSIVE ROTATION (Conceptual Beamforming)

I1 = real(fft(pnor) .* conj(fft(vxnor)));

I2 = real(fft(pnor) .* conj(fft(vynor)));

theta = 360 * atan(sum(I2)/sum(I1)) / (2*pi); % degrees

vc = vxnor * cosd(theta) + vynor * sind(theta);

resultant_passive = pnor + 2*vc;

%% ELECTRONICALLY STEERED BEAM

theta_steer = 60; % Desired steering angle in degrees

lambda = 1; % Normalized wavelength

d = 0.5*lambda; % Antenna spacing

% Phase shifts for each antenna

phi_vx = -2*pi*d/lambda * cosd(theta_steer);

phi_vy = -2*pi*d/lambda * cosd(theta_steer);

% Apply phase shifts (complex representation)

vx_steered = vxnor .* cos(phi_vx) - vxnor .* sin(phi_vx)*1j;

vy_steered = vynor .* cos(phi_vy) - vynor .* sin(phi_vy)*1j;

% Coherent sum and take real part for visualization

resultant_steered = real(pnor + vx_steered + vy_steered);

%% PLOT VECTORS IN 2D

figure('Color','w'); hold on; grid on; axis equal;

xlabel('vx'); ylabel('vy'); title('2D Beamforming: Passive vs Electronic Steering');

sample_idx = 200; % Sample to illustrate

% Original antenna vectors

quiver(0,0,vxnor(sample_idx),0,0,'r','LineWidth',2,'MaxHeadSize',0.5); % vx

quiver(0,0,0,vynor(sample_idx),0,'b','LineWidth',2,'MaxHeadSize',0.5); % vy

quiver(0,0,pnor(sample_idx)*cosd(45), pnor(sample_idx)*sind(45),0,'k','LineWidth',2,'MaxHeadSize',0.5); % reference

% Passive rotated vector

quiver(0,0,vc(sample_idx)*cosd(theta), vc(sample_idx)*sind(theta),0,'m','LineWidth',2,'MaxHeadSize',0.5);

%% ELECTRONICALLY STEERED BEAM (corrected)

theta_steer = 120; % Desired steering angle in degrees

lambda = 1; % Normalized wavelength

d = 0.5*lambda; % Antenna spacing

% Complex phase shifts for each antenna element (assuming linear array along x)

phi_vx = -2*pi*d/lambda * cosd(theta_steer);

phi_vy = -2*pi*d/lambda * cosd(theta_steer);

% Antenna phasors at the sample index (using normalized amplitudes and zero phase for simplicity)

vx_phasor = vxnor(sample_idx) * exp(1j*phi_vx);

vy_phasor = vynor(sample_idx) * exp(1j*phi_vy);

p_phasor = pnor(sample_idx); % reference antenna assumed at phase 0

% Sum phasors coherently

resultant_phasor = p_phasor + vx_phasor + vy_phasor;

% Plot the electronically steered beam as a 2D vector (real vs imag parts)

quiver(0, 0, real(resultant_phasor), imag(resultant_phasor), 0, ...

'g', 'LineWidth', 2, 'MaxHeadSize', 0.5);

% Annotate

legend('vx','vy','Reference p','Passive Rotation vc','Electronic Steering','Location','best');

% Show theta arcs

theta_rad_passive = deg2rad(theta);

r_arc = 0.4;

arc_x = r_arc*cos(linspace(0,theta_rad_passive,50));

arc_y = r_arc*sin(linspace(0,theta_rad_passive,50));

plot(arc_x, arc_y, 'm--','LineWidth',1.5);

text(0.15,0.05,['\theta_{passive} = ',num2str(theta,'%.2f'),'°'],'FontSize',12,'Color','m');

theta_rad_steer = deg2rad(theta_steer);

arc_x2 = r_arc*cos(linspace(0,theta_rad_steer,50));

arc_y2 = r_arc*sin(linspace(0,theta_rad_steer,50));

plot(arc_x2, arc_y2, 'g--','LineWidth',1.5);

text(0.15,0.1,['\theta_{steer} = ',num2str(theta_steer,'%.2f'),'°'],'FontSize',12,'Color','g');

xlim([-1 1]); ylim([-1 1]);

grid on;

 Output

 

Workflow of 2D Conceptual Beamforming Code

1. Signal Simulation

Simulate sinusoidal signals received by three antennas with phase offsets:

Reference signal (antenna p):

\( p(t) = \sin(2\pi f t) \)

Antennas vx and vy with phase shifts:

\( v_x(t) = A_x \sin(2\pi f t + \phi_x) \)

\( v_y(t) = A_y \sin(2\pi f t + \phi_y) \)

2. Normalization

Normalize signals to scale amplitude between -1 and 1:

pnor = p / max(abs(p));
vxnor = vx / max(abs(vx));
vynor = vy / max(abs(vy));

3. Cross-Spectrum Computation

Compute the cross-spectrum terms between the reference and antenna signals in frequency domain:

Let \( P(f) = \mathrm{FFT}[p_{nor}(t)] \), \( V_x(f) = \mathrm{FFT}[v_{x,nor}(t)] \), and \( V_y(f) = \mathrm{FFT}[v_{y,nor}(t)] \).

Cross-spectra are:

\[ I_1(f) = \Re \{ P(f) \cdot V_x^*(f) \} \]

\[ I_2(f) = \Re \{ P(f) \cdot V_y^*(f) \} \]

where \( ^* \) denotes complex conjugate and \( \Re \{ \cdot \} \) the real part.

4. Estimated Rotation Angle (Beam Direction)

Estimate the beam rotation angle \( \theta \) from the sum of the cross-spectrum components:

\[ \theta = \frac{360}{2\pi} \arctan\left( \frac{\sum_f I_2(f)}{\sum_f I_1(f)} \right) \]

This angle estimates the direction of arrival (DoA) or rotation needed to align the signals.

5. Beamforming Combination

Rotate antenna components using the estimated angle and combine signals for conceptual beamforming:

\[ v_c(t) = v_{x,nor}(t) \cos \theta + v_{y,nor}(t) \sin \theta \]

\[ \mathrm{Resultant}(t) = p_{nor}(t) + 2 v_c(t) \]

6. Visualization

• Plot individual antenna signals and the combined beamformed output in the time domain.
• Optionally plot frequency spectra and vector plots for conceptual illustration.

Summary

This workflow uses frequency-domain cross-spectra to estimate relative phases and direction, then applies rotation to combine signals coherently — the essence of beamforming.